Towards Morphological System Design Inst. for Information Transmission Problems Russian Academy of Sciences Mark Sh. Levin INDIN 2009: “IEEE Conf. on Industrial Informatics”: Cardiff Univ., June 2009 PLAN: 1.Introduction: System configuration problem (selection) 2.Morphological analysis (F. Zwicky, 1943) 3.Ideal point method (A. Ayres & V. Iakimets, 1969 …) 4.Morphological analysis and optimization, multiple choice problem (1986…) 5.Pareto-based approach (RAS, ) 6.HMMD (M.Sh. Levin, 1994) 7.Allocation of system components 8.HMMD with uncertainty (M.Sh. Levin, 1998) 9.Conclusion

System (parts/components): S = P(1) * … * P(i) * … * P(m) Configuration example: S 1 = X 1 2 * … * X i 1 * … * X m 3 P(1) X 1 1 … X 1 q1 P(m) X m 1 … X m q1 P(i) X i 1 … X i q1 This is about problem of representatives, multiple choice problem SYSTEM CONFIGURATION: SELECTION OF SYSTEM COMPONENTS

BASIC REFERENCES FOR SYSTEM CONFIGURATION 1.M.Sh. Levin, Combinatorial Optimization in Systems Configuration Design, Automation & Remote Control (Springer), 70(3), pp , Configuration design in AI (a set of papers and several workshops; versions of SAT problems)

MORPHOLOGICAL ANALYSIS (Zwicky, 1943) A1A1 AiAi AnAn System... Morphological class 1: |A 1 | = m(1) Morphological class i: |A i | = m(i) Morphological class n: |A n | = m(n) Subsystem 1 Subsystem nSubsystem i

BASIC REFERENCES 1.F. Zwicky, Discovery Invention: Research Through the Morphological Analysis. New York, McMillan, R.Y. Ayres, Technological Forecasting and Long-Time Planning, New York, McGraw-Hill, J.C. Jones, Design Methods. 2 nd ed., New York, Wiley, T. Ritchey, Problem structuring using computer-aided morphological analysis J. of Operational Research Society, Vol. 57, no. 7, pp , 2006.

IDEAL POINT METHOD (Closeness to “IDEAL”) (Ayres, 1969) X Y Z System... Subsystem X Subsystem ZSubsystem Y X1X1 ID i ID n ID 1 X2X2 X3X3 X4X4 Y1Y1 Y2Y2 Y3Y3 Y4Y4 Y5Y5 Z1Z1 Z2Z2 Z3Z3 “IDEAL” : X 1 * Y 3 * Z 3 (infeasible combination) S 1 = X 4 * Y 3 * Z 3 S 2 = X 2 * Y 5 * Z 1  (“IDEAL”, S 1 ) <  (“IDEAL”, S 2 )  is a closeness (proximity)

BASIC REFERENCES 1.R.Y. Ayres, Technological Forecasting and Long-Time Planning, New York, McGraw-Hill, Yu.A. Dubov, S.I. Travkin, V.N. Iakimets, Multicriteria Models for Building of System Choice Variants. Moscow, Nauka, 1986 (in Russian).

Multicriteria evaluation of feasible combinations & selection of Pareto-effective points ( ) X Y Z System... Subsystem X Subsystem ZSubsystem Y X1X1 X2X2 X3X3 X4X4 Y1Y1 Y2Y2 Y3Y3 Y4Y4 Y5Y5 Z1Z1 Z2Z2 Z3Z3 STEP 1. Generation of feasible combinations: S 3 = X 4 * Y 2 * Z 3 S 4 = X 4 * Y 1 * Z 3 S 1 = X 4 * Y 3 * Z 3 S 2 = X 2 * Y 5 * Z 1 STEP 2. Evaluation upon criteria STEP 3. Selection of Pareto-effective solutions

BASIC REFERENCES 1.S.V. Emeljanov, V.M. Ozernoy, O.I. Larichev, e al., Choice of rational variants of technological mines upon multiple criteria, Mining J., no. 5, 1972 (in Russian). 2.M.G. Gaft, N.N. Milovidov, V.I. Serov, E.D. Gusev, Decision making method for choice of rational configurations of cars. Problems&Methods of Decision Making in Organizational Management Systems. Moscow. Inst. for Systems Analysis, pp , 1982 (in Russian)

max  m i=1  qi j=1 c ij x ij s.t.  m i=1  qi j=1 a ij x ij  b  qi j=1 x ij  1, i = 1, …, m x ij  {0, 1}, i = 1, …, m, j = 1, …, qi... J 1 J i J m...  i | J i | = qi, j = 1, …, qi SELECTION OF SYSTEM COMPONENTS: MULTIPLE CHOICE PROBLEM Applications of multiple choice problem: (i)Software systems (ii)Hardware (ii)Improvement of communication systems

Integer nonlinear programming (modular design of series system from the viewpoint of reliability) (by Berman&Ashrafi) max  m i=1 (1 -  qi j=1 ( 1 - p ij x ij ) ) s.t.  m i=1  qi j=1 d ij x ij  b  qi j=1 x ij  1, i = 1, …, m x ij  {0, 1}, i = 1, …, m, j = 1, …, qi p ij is reliability, d ij is cost... J 1 J i J m...  i | J i | = qi, j = 1, …, qi 

BASIC REFERENCES FOR SYSTEM CONFIGURATION 1.M.R. Garey, D.S. Johnson, Computers and Intractability. Freeman, S. Martello, P. Toth, Knapsack Problem, New York, Wiley, H. Kellerer, U. Pferschy, D. Pisinger, Knapsack Problem. Springer, O. Berman, N. Ashrafi, Optimization Models for Reliable Software System. IEEE Trans. Software Eng., 19(11), pp , V. Poladian et. al., Task-Base adaptation for ubiquitous computing. IEEE Trans. SMC, 36(3), pp , M.Sh. Levin, A.V. Safonov, Design and redesign of configuration for facility in communication network. Inform. Technol. & Comp. Systems, Issue 4, pp , 2006 (in Russian) 7.M.Sh.Levin, Combinatorial Optimization in Systems Configuration Design, Autom.&Rem. Control (Springer), 70(3), pp , 2009.

MORPHOLOGICAL CLIQUE (or multipartite clique) PART 1 PART 2 PART 3 Vertices (design alternatives) Edges (compatibility) NOTE: about k-matching

Hierarchical Morphological Multicriteria Design (HMMD) ZXY X 1 (2) X 2 (1) X 3 (1) Z 1 (1) Z 2 (1) Z 3 (2) Y 1 (3) Y 2 (1) Y 3 (2) Y 4 (3) S=X*Y*Z S 1 =X 1 *Y 4 *Z 3

Concentric presentation of morphological clique with estimates of compatibility X 3, X 2 X1X1 Z3Z3 Z 1, Z 2 Y4Y1Y4Y1 Y3Y3 Y2Y

Two-criteria space of quality for resultant combinations Ideal Point Quality of compatibility Quality of elements Pareto-effective points (combinations)

Discrete space of composite system quality (by components) S1S1 The Ideal Point The Worst Point

Discrete space of composite system quality (by components, by compatibility) Ideal Point w=1 N(S 1 ) w=3 w=2 THIS IS THE SPACE OF VECTORS: N(S) = ( w (S) ; n 1 (S), n 2 (S), n 3 (S) ) where w (S) is the estimate of compatibility (e.g., minimum of pairwise compatibility estimates) & n 1 (S) is the number of components at the 1 st quality level, etc.

Two types of solving schemes 1.Enumerative directed heuristic: analysis and checking since the best point. 2.Dynamic programming methods: extension of the method for knapsack problem or multiple choice problem.

S=X*Y*Z Y Z=P*Q*U*V Z 1 =P 2 *Q 3 *U 1 *V 5 Z 2 =P 1 *Q 2 *U 3 *V 1 Y1Y2Y3Y1Y2Y3 AB A1A2A3A1A2A3 B1B2B3B4B1B2B3B4 C1C2C3C4C5C1C2C3C4C5 D=I*J I1I2I3I1I2I3 J1J2J3J4J1J2J3J4 P1P2P3P1P2P3 Q1Q2Q3Q4Q1Q2Q3Q4 U1U2U3U1U2U3 V1V2V3V4V5V6V1V2V3V4V5V6 C X=A*B*C*D P QU V X 1 =A 1 *B 2 *C 4 *D 3 X 2 =A 3 *B 4 *C 2 *D 1 D 1 =I 1 *J 1 D 2 =I 1 *J 2 D 3 =I 3 *J 4 S 1 =X 2 *Y 3 *Z 2 S 2 =X 1 *Y 2 *Z 1 J I Illustration for HMMD (based on morphological clique problem) Compatibility for I,J Compatibility for P,Q,U,V Compatibility for A,B,C,D Compatibility for X,Y,Z

BASIC REFERENCES 1.M.Sh. Levin, Combinatorial Engineering of Decomposable Systems. Boston/Dordrecht, Kluwer, M.Sh. Levin, Composite Systems Decisions, London/New York, Springer, M.Sh. Levin, Papers 4.M.Sh. Levin, Educational Courses (& Research with Students): “System Design” (recent course in Moscow Inst. of Physics &Technology, 2004…2008), etc.

ASSIGNMENT/ALLOCATION Allocation (assignment, matching, location): BIPARTITE GRAPH a b c d e f g h Positions (locations, sites) Set of elements (e.g., personnel, facilities) MAPPING

ASSIGNMENT as morphology... a b cd Positions (locations, sites) Candi- dates for position a Candi- dates for position d Candi- dates for position c Candi- dates for position b...

BASIC REFERENCES FOR ASSIGNMENT/ALLOCATION 1.M.R. Garey, D.S. Johnson, Computers and Intractability. New York, Freeman, E. Cela, The Quadratic Assignment Problem. Kluwer, P.M. Pardalos, H. Wolkowicz, (Eds.), Quadratic Assignment and Related Problems, Providence, AMS, M.Sh. Levin, Combinatorial Engineering of Decomposable Systems. Kluwer, M.Sh. Levin, Combinatorial optimization in system configuration design. Automation & Remote (Springer), 70(3), pp , 2009.

Hierarchical Morphological Multicriteria Design (HMMD) with Uncertainty Uncertainty: 1.Probabilistic estimates 2.Fuzzy estimates 3.etc. A.Estimates of DAs: (0)deterministic (1)probabilistic (2)fuzzy estimates (e.g., interval estimates) B.Estimates of compatibility: (0)deterministic (1)probabilistic (1)fuzzy estimates (e.g., interval estimates) RESULT: 9 CASES

Hierarchical Morphological Multicriteria Design (HMMD) with Uncertainty METHODOLOGICAL APPROACHES: Case 1 (Basic): DAs: deterministic (aggregated) estimates Compatibility: deterministic (aggregated) estimates Case 2. DAs: deterministic (aggregated) estimates Compatibility: fuzzy estimates Case 3. DAs: fuzzy estimates Compatibility: deterministic (aggregated) estimates Case 4. DAs: fuzzy estimates Compatibility: fuzzy estimates

Two-criteria space of quality for resultant combinations Ideal Point Quality of compatibility Quality of elements Case 1 Case 2 Case 3 Case 4

BASIC REFERENCES 1.M.Sh. Levin, Combinatorial Engineering of Decomposable systems, Kluwer, M.Sh. Levin, Papers in progress